Causal conformal vector fields,and singularities of twistor spinors

In this paper, we study the geometry around the singularity of a twistor spinor, on a Lorentz manifold (M, g) of dimension greater or equal to three, endowed with a spin structure. Using the dynamical properties of conformal vector fields, we prove that the geometry has to be conformally flat on some open subset of any neighbourhood of the singularity. As a consequence, any analytic Lorentz manifold, admitting a twistor spinor with at least one zero has to be conformally flat.      

Non--trivial harmonic spinors on generic algebraic surfaces

We show that there are simply connected spin algebraic surfaces for which all complex structures in certain components of the moduli space admit more harmonic spinors than predicted by the index theorem (or Riemann--Roch). The dimension of the space of harmonic spinors can exceed the absolute value of the index by an arbitrarily large number.

     

Canonical Lorentzian spin structure and twistor spinors on the Fefferman space of a contact Riemannian manifold

The Fefferman space of a contact Riemannian manifold carries a Lorentzian spin structure canonically. On the Lorentzian spin manifold, we investigate the Dirac operator and the twistor operator closely. In particular, we show that, if the contact Riemannian manifold is integrable, then there exist non-zero global solutions of the twistor equation.     

The super-Toda system and bubbling of spinors

We introduce the super-Toda system on Riemann surfaces and study the blow-up analysis for a sequence of solutions to the super-Toda system on a closed Riemann surface with uniformly bounded energy. In particular, we show the energy identities for the spinor parts of a blow-up sequence of solutions for which there are possibly four types of bubbling solutions, namely, finite energy solutions of the super-Liouville equation or the super-Toda system defined on ${\mathbb{R}}^2$ or on ${\mathbb{R}}^2?\{0\}$. This is achieved by showing some new energy gap results for the spinor parts of these four types of bubbling solutions.     

Concomitants and syzygies of spinors in 3-dimensional euclidean space

Summary In 3-dimensional Euclidean space, explicit formulae for the Hermitean concomitants of two irreducible spinsors of type[n+1/2] are obtained, where n is a positive integer. Further, a method, by means of which the syzygies of degree2 in these concomitants can be obtained, is described and is illustrated by considering two irreducible spinors of type[3/2]. Entrata in Redazione il 16 maggio 1972.     

The Dirac operator on symplectic spinors

Symplectic spinors were introduced by B. Kostant in [4] in the context of geometric quantization. This paper presents further considerations concerning symplectic spinors. We define the spinor derivative induced by a symplectic covariant derivative. We compute an explicit formula for this spinor derivative and prove some elementary properties. This makes it possible to define the symplectic Dirac operator in a canonical way. In case of a symplectic and torsion-free covariant derivative it turns out to be formally selfadjoint.     

The twistor structure of the biquaternionic projective point

Souček [1, 2] discovered an intriguing connection between the standard twistor correspondence and the biquaternionic projective line The biquaternionic projective point, also has twistor structure corresponding to the collection of α- or β-planes passing through the origin in spacetime. The duality between α- or β-planes is shown to correspond to the choice of left vs. right scalar action. Moreover, we find that is homeomorphic to the scheme      

The length function of a twistor spinor with zero

We examine the Taylor expansion of the length function of a twistor spinor with zero on a Riemannian orbifold around its zero and study it on the Eguchi–Hanson orbifold. This expansion is written in some conformal normal coordinates (CNC) around the zero up to order 7. In the example of the Eguchi–Hanson orbifold, CNC are found explicitly. We use the expansion in computing the mass (a generalization of ADM–mass) of the asymptotically locally Euclidean coordinate system, which is constructed from a conformal normal coordinate system around the zero of a twistor spinor on a Riemannian spin orbifold admitting isolated singularities.     

Generalized metrics and generalized twistor spaces

Mathematische Zeitschrift - The twistor construction for Riemannian manifolds is extended to the case of manifolds endowed with generalized metrics (in the sense of generalized geometry à la...     

The center of the ring of 3×3 generic matrices

Following Procesi, the center of the division ring of generic matrices over a field F is described as the fixed field of the symmetric group acting on a purely transcendental extension of F. For 3×3 matrices, the center is shown to be purely transcendental over F. In characteristic zero this is equivalent to saying that the field of simultaneous rational invariants of 3×3 matrices over F is a purely transcendental extension field of F.     

Slice-polynomial functions and twistor geometry of ruled surfaces in $$mathbb {CP}^3$$ CP 3

Mathematische Zeitschrift - In the present paper we introduce the class of slice-polynomial functions: slice regular functions defined over the quaternions, outside the real axis, whose restriction...     

The first Chern class and conformal area for a twistor holomorphic immersion

We obtain an inequality involving the first Chern class of the normal bundle and the conformal area for a twistor holomorphic surface. Using this inequality, we can improve an inequality obtained by T. Friedrich for the Euler class of the normal bundle of a twistor holomorphic surface in the four-dimensional space form. Moreover, as a corollary, we see that the area of a superminimal surface in the unit sphere is an integer multiple of $2 \pi $ , which is essentially proved by E. Calabi.     

The non-solvability by radicals of generic 3-connected planar Laman graphs

We show that planar embeddable -connected Laman graphs are generically non-soluble. A Laman graph represents a configuration of points on the Euclidean plane with just enough distance specifications between them to ensure rigidity. Formally, a Laman graph is a maximally independent graph, that is, one that satisfies the vertex-edge count together with a corresponding inequality for each subgraph. The following main theorem of the paper resolves a conjecture of Owen (1991) in the planar case. Let be a maximally independent -connected planar graph, with more than 3 vertices, together with a realisable assignment of generic distances for the edges which includes a normalised unit length (base) edge. Then, for any solution configuration for these distances on a plane, with the base edge vertices placed at rational points, not all coordinates of the vertices lie in a radical extension of the distance field.

     

The 1-2-3-conjecture holds for dense graphs

This paper confirms the 1-2-3-conjecture for graphs that can be edge-decomposed into cliques of order at least 3. Furthermore we combine this with a result by Barber, Kühn, Lo, and Osthus to show that there is a constants such that every graph with and , where is the minimum degree of satisfying the 1-2-3-conjecture.     

The scalar curvature equation on 2- and 3-spheres

In this article we obtain a priori estimates for solutions to the prescribed scalar curvature equation on 2- and 3-spheres under a nondegeneracy assumption on the curvature function. Using this estimate, we use the continuity method to demonstrate the existence of solutions to this equation when a map associated to the given curvature function has non-zero degree.Research of first author supported in part by NSF grant 91-03949Research of second author supported by a NSF Postdoctoral Fellowship.Research of third author supported in part by NSF grant 91-02872 and the Ellentuck Fund.